Rotational motion around a fixed axis is another special case of rigid body motion.
Rotational movement of a rigid body around a fixed axis
it is called such a movement in which all points of the body describe circles, the centers of which are on the same straight line, called the axis of rotation, while the planes to which these circles belong are perpendicular rotation axis
(Fig.2.4).
In technology, this type of motion occurs very often: for example, the rotation of the shafts of engines and generators, turbines and aircraft propellers.
Angular velocity
. Each point of a body rotating around an axis passing through the point ABOUT, moves in a circle, and different points travel different paths over time. So, , therefore the modulus of the point velocity A more than a point IN (Fig.2.5). But the radii of the circles rotate through the same angle over time. Angle - the angle between the axis OH and radius vector, which determines the position of point A (see Fig. 2.5).
Let the body rotate uniformly, i.e., rotate through equal angles at any equal intervals of time. The speed of rotation of a body depends on the angle of rotation of the radius vector, which determines the position of one of the points of the rigid body for a given period of time; it is characterized angular velocity
.
For example, if one body rotates through an angle every second, and the other through an angle, then we say that the first body rotates 2 times faster than the second.
Angular velocity of a body during uniform rotation
is a quantity equal to the ratio of the angle of rotation of the body to the period of time during which this rotation occurred.
We will denote the angular velocity by the Greek letter ω
(omega). Then by definition
Angular velocity is expressed in radians per second (rad/s).
For example, the angular velocity of the Earth's rotation around its axis is 0.0000727 rad/s, and that of a grinding disk is about 140 rad/s 1 .
Angular velocity can be expressed through rotation speed
, i.e. the number of full revolutions in 1s. If a body makes (Greek letter “nu”) revolutions in 1s, then the time of one revolution is equal to seconds. This time is called rotation period
and denoted by the letter T. Thus, the relationship between frequency and rotation period can be represented as:
A complete rotation of the body corresponds to an angle. Therefore, according to formula (2.1)
If during uniform rotation the angular velocity is known and at the initial moment of time the angle of rotation is , then the angle of rotation of the body during time t according to equation (2.1) is equal to:
If , then , or 
.
Angular velocity takes positive values if the angle between the radius vector, which determines the position of one of the points of the rigid body, and the axis OH increases, and negative when it decreases.
Thus, we can describe the position of the points of a rotating body at any time.
Relationship between linear and angular velocities.
The speed of a point moving in a circle is often called linear speed
, to emphasize its difference from angular velocity.
We have already noted that when a rigid body rotates, its different points have unequal linear velocities, but the angular velocity is the same for all points.
There is a relationship between the linear speed of any point of a rotating body and its angular speed. Let's install it. A point lying on a circle of radius R, will cover the distance in one revolution. Since the time of one revolution of a body is a period T, then the modulus of the linear velocity of the point can be found as follows:
When describing the movement of a point along a circle, we will characterize the movement of the point by the angle Δφ , which describes the radius vector of a point over time Δt. Angular displacement in an infinitesimal period of time dt denoted by dφ.
Angular displacement is a vector quantity. The direction of the vector (or ) is determined by the gimlet rule: if you rotate the gimlet (screw with a right-hand thread) in the direction of the point’s movement, the gimlet will move in the direction of the angular displacement vector. In Fig. 14 point M moves clockwise if you look at the plane of movement from below. If you twist the gimlet in this direction, the vector will be directed upward.
Thus, the direction of the angular displacement vector is determined by the choice of the positive direction of rotation. The positive direction of rotation is determined by the right-hand thread gimlet rule. However, with the same success one could take a gimlet with a left-hand thread. In this case, the direction of the angular displacement vector would be opposite.
When considering such quantities as speed, acceleration, displacement vector, the question of choosing their direction did not arise: it was determined naturally from the nature of the quantities themselves. Such vectors are called polar. Vectors similar to the angular displacement vector are called axial, or pseudovectors. The direction of the axial vector is determined by choosing the positive direction of rotation. In addition, the axial vector does not have an application point. Polar vectors, which we have considered so far, are applied to a moving point. For an axial vector, you can only indicate the direction (axis, axis - Latin) along which it is directed. The axis along which the angular displacement vector is directed is perpendicular to the plane of rotation. Typically, the angular displacement vector is drawn on an axis passing through the center of the circle (Fig. 14), although it can be drawn anywhere, including on an axis passing through the point in question.
In the SI system, angles are measured in radians. A radian is an angle whose arc length is equal to the radius of the circle. Thus, the total angle (360 0) is 2π radians.
Motion of a point in a circle
Angular velocity– vector quantity, numerically equal to the angle of rotation per unit time. Angular velocity is usually denoted by the Greek letter ω. By definition, angular velocity is the derivative of an angle with respect to time:
. (19)
The direction of the angular velocity vector coincides with the direction of the angular displacement vector (Fig. 14). The angular velocity vector, just like the angular displacement vector, is an axial vector.
The dimension of angular velocity is rad/s.
Rotation with a constant angular velocity is called uniform, with ω = φ/t.
Uniform rotation can be characterized by the rotation period T, which is understood as the time during which the body makes one revolution, i.e., rotates through an angle of 2π. Since the time interval Δt = T corresponds to the rotation angle Δφ = 2π, then
(20)
The number of revolutions per unit time ν is obviously equal to:
(21)
The value of ν is measured in hertz (Hz). One hertz is one revolution per second, or 2π rad/s.
The concepts of the period of revolution and the number of revolutions per unit time can also be preserved for non-uniform rotation, understanding by the instantaneous value T the time during which the body would make one revolution if it rotated uniformly with a given instantaneous value of angular velocity, and by ν meaning that number revolutions that a body would make per unit time under similar conditions.
If the angular velocity changes with time, then the rotation is called uneven. In this case enter angular acceleration in the same way as linear acceleration was introduced for rectilinear motion. Angular acceleration is the change in angular velocity per unit time, calculated as the derivative of angular velocity with respect to time or the second derivative of angular displacement with respect to time:
(22)
Just like angular velocity, angular acceleration is a vector quantity. The angular acceleration vector is an axial vector, in the case of accelerated rotation it is directed in the same direction as the angular velocity vector (Fig. 14); in the case of slow rotation, the angular acceleration vector is directed opposite to the angular velocity vector.
With uniformly variable rotational motion, relations similar to formulas (10) and (11), which describe uniformly variable rectilinear motion, take place:
ω = ω 0 ± εt,
.
Circular motion is a special case of curvilinear motion. The speed of a body at any point of a curvilinear trajectory is directed tangentially to it (Fig. 2.1). In this case, the speed as a vector can change both in magnitude (magnitude) and direction. If the speed module 
remains unchanged, then we talk about uniform curvilinear motion.
Let a body move in a circle with a constant speed from point 1 to point 2.
In this case, the body will travel a path equal to the length of the arc ℓ 12 between points 1 and 2 in time t. During the same time, the radius vector R drawn from the center of the circle 0 to the point will rotate through an angle Δφ.
The velocity vector at point 2 differs from the velocity vector at point 1 by direction by the value ΔV:
;
To characterize the change in the velocity vector by the value δv, we introduce acceleration:
(2.4)
Vector 
at any point of the trajectory directed along the radius Rк center circle perpendicular to the velocity vector V 2. Therefore the acceleration 
, which characterizes the change in speed during curvilinear movement 
in direction is called centripetal or normal. Thus, the movement of a point along a circle with a constant absolute speed is accelerated.
If the speed 
changes not only in direction, but also in modulus (magnitude), then in addition to normal acceleration 
they also introduce tangent (tangential) acceleration 
, which characterizes the change in speed in magnitude:
or 
Directed vector 
along a tangent at any point of the trajectory (i.e. coincides with the direction of the vector 
). Angle between vectors 
And 
equals 90 0.
The total acceleration of a point moving along a curved path is defined as a vector sum (Fig. 2.1.).
.
Vector module 
.
Angular velocity and angular acceleration
When a material point moves circumferentially The radius vector R, drawn from the center of the circle O to the point, rotates through an angle Δφ (Fig. 2.1). To characterize rotation, the concepts of angular velocity ω and angular acceleration ε are introduced.
The angle φ can be measured in radians. 1 rad is equal to the angle that rests on the arc ℓ equal to the radius R of the circle, i.e.
or ℓ
12
=
Rφ
(2.5.)
Let us differentiate equation (2.5.)
(2.6.)
Value dℓ/dt=V instant. The quantity ω =dφ/dt is called angular velocity(measured in rad/s). Let us obtain the relationship between linear and angular velocities:
The quantity ω is vector. Vector direction 
determined screw rule: it coincides with the direction of movement of the screw, oriented along the axis of rotation of a point or body and rotated in the direction of rotation of the body (Fig. 2.2), i.e. 
.
Angular acceleration
called the vector quantity derivative of the angular velocity (instantaneous angular acceleration)
,
(2.8.)
Vector 
coincides with the axis of rotation and is directed in the same direction as the vector 
, if the rotation is accelerated, and in the opposite direction if the rotation is slow.
Speednbodies per unit time are calledrotation speed .
The time T for one full revolution of the body is calledrotation period . WhereinRdescribes the angle Δφ=2π radians
With that said
,
(2.9)
Equation (2.8) can be written as follows:
(2.10)
Then the tangential component of acceleration
and =R(2.11)
Normal acceleration a n can be expressed as follows:
taking into account (2.7) and (2.9)
(2.12)
Then full acceleration.
For rotational motion with constant angular acceleration , we can write the kinematics equation by analogy with equation (2.1) – (2.3) for translational motion:
,
.
1.Uniform movement in a circle
2. Angular speed of rotational motion.
3. Rotation period.
4. Rotation speed.
5. Relationship between linear speed and angular speed.
6.Centripetal acceleration.
7. Equally alternating movement in a circle.
8. Angular acceleration in uniform circular motion.
9.Tangential acceleration.
10. Law of uniformly accelerated motion in a circle.
11. Average angular velocity in uniformly accelerated motion in a circle.
12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.
1.Uniform movement around a circle– movement in which a material point passes equal segments of a circular arc in equal time intervals, i.e. the point moves in a circle with a constant absolute speed. In this case, the speed is equal to the ratio of the arc of a circle traversed by the point to the time of movement, i.e.
and is called the linear speed of movement in a circle.
As in curvilinear motion, the velocity vector is directed tangentially to the circle in the direction of motion (Fig. 25).
2. Angular velocity in uniform circular motion– ratio of the radius rotation angle to the rotation time:
In uniform circular motion, the angular velocity is constant. In the SI system, angular velocity is measured in (rad/s). One radian - a rad is the central angle subtending an arc of a circle with a length equal to the radius. A full angle contains radians, i.e. per revolution the radius rotates by an angle of radians.
3. Rotation period– time interval T during which a material point makes one full revolution. In the SI system, the period is measured in seconds.
4. Rotation frequency– the number of revolutions made in one second. In the SI system, frequency is measured in hertz (1Hz = 1). One hertz is the frequency at which one revolution is completed in one second. It's easy to imagine that
If during time t a point makes n revolutions around a circle then .
Knowing the period and frequency of rotation, the angular velocity can be calculated using the formula:
5 Relationship between linear speed and angular speed. The length of an arc of a circle is equal to where is the central angle, expressed in radians, the radius of the circle subtending the arc. Now we write the linear speed in the form
It is often convenient to use the formulas: or Angular velocity is often called cyclic frequency, and frequency is called linear frequency.
6. Centripetal acceleration. In uniform motion around a circle, the velocity module remains unchanged, but its direction continuously changes (Fig. 26). This means that a body moving uniformly in a circle experiences acceleration, which is directed towards the center and is called centripetal acceleration.
Let a distance travel equal to an arc of a circle in a period of time. Let's move the vector, leaving it parallel to itself, so that its beginning coincides with the beginning of the vector at point B. The modulus of change in speed is equal to , and the modulus of centripetal acceleration is equal
In Fig. 26, the triangles AOB and DVS are isosceles and the angles at the vertices O and B are equal, as are the angles with mutually perpendicular sides AO and OB. This means that the triangles AOB and DVS are similar. Therefore, if, that is, the time interval takes arbitrarily small values, then the arc can be approximately considered equal to the chord AB, i.e. . Therefore, we can write Considering that VD = , OA = R we obtain Multiplying both sides of the last equality by , we further obtain the expression for the modulus of centripetal acceleration in uniform motion in a circle: . Considering that we get two frequently used formulas:
So, in uniform motion around a circle, the centripetal acceleration is constant in magnitude.
It is easy to understand that in the limit at , angle . This means that the angles at the base of the DS of the ICE triangle tend to the value , and the speed change vector becomes perpendicular to the speed vector, i.e. directed radially towards the center of the circle.
7. Equally alternating circular motion– circular motion in which the angular velocity changes by the same amount over equal time intervals.
8. Angular acceleration in uniform circular motion– the ratio of the change in angular velocity to the time interval during which this change occurred, i.e.
where the initial value of angular velocity, the final value of angular velocity, angular acceleration, in the SI system is measured in . From the last equality we obtain formulas for calculating the angular velocity
And if .
Multiplying both sides of these equalities by and taking into account that , is the tangential acceleration, i.e. acceleration directed tangentially to the circle, we obtain formulas for calculating linear speed:
And if .
9. Tangential acceleration numerically equal to the change in speed per unit time and directed along the tangent to the circle. If >0, >0, then the motion is uniformly accelerated. If<0 и <0 – движение.
10. Law of uniformly accelerated motion in a circle. The path traveled around a circle in time in uniformly accelerated motion is calculated by the formula:
Substituting , , and reducing by , we obtain the law of uniformly accelerated motion in a circle:
Or if.
If the movement is uniformly slow, i.e.<0, то
11.Total acceleration in uniformly accelerated circular motion. In uniformly accelerated motion in a circle, centripetal acceleration increases over time, because Due to tangential acceleration, linear speed increases. Very often, centripetal acceleration is called normal and is denoted as. Since the total acceleration at a given moment is determined by the Pythagorean theorem (Fig. 27).
12. Average angular velocity in uniformly accelerated motion in a circle. The average linear speed in uniformly accelerated motion in a circle is equal to . Substituting here and and reducing by we get
If, then.
12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.
Substituting the quantities , , , , into the formula
and reducing by , we get
Lecture-4. Dynamics.
1. Dynamics
2. Interaction of bodies.
3. Inertia. The principle of inertia.
4. Newton's first law.
5. Free material point.
6. Inertial reference system.
7. Non-inertial reference system.
8. Galileo's principle of relativity.
9. Galilean transformations.
11. Addition of forces.
13. Density of substances.
14. Center of mass.
15. Newton's second law.
16. Unit of force.
17. Newton's third law
1. Dynamics there is a branch of mechanics that studies mechanical motion, depending on the forces that cause a change in this motion.
2.Interactions of bodies. Bodies can interact both in direct contact and at a distance through a special type of matter called a physical field.
For example, all bodies are attracted to each other and this attraction is carried out through the gravitational field, and the forces of attraction are called gravitational.
Bodies carrying an electric charge interact through an electric field. Electric currents interact through a magnetic field. These forces are called electromagnetic.
Elementary particles interact through nuclear fields and these forces are called nuclear.
3.Inertia. In the 4th century. BC e. The Greek philosopher Aristotle argued that the cause of the movement of a body is the force acting from another body or bodies. At the same time, according to Aristotle’s movement, a constant force imparts a constant speed to the body and, with the cessation of the action of the force, the movement ceases.
In the 16th century Italian physicist Galileo Galilei, conducting experiments with bodies rolling down an inclined plane and with falling bodies, showed that a constant force (in this case, the weight of a body) imparts acceleration to the body.
So, based on experiments, Galileo showed that force is the cause of the acceleration of bodies. Let us present Galileo's reasoning. Let a very smooth ball roll along a smooth horizontal plane. If nothing interferes with the ball, then it can roll for as long as desired. If a thin layer of sand is poured on the path of the ball, it will stop very soon, because it was affected by the frictional force of the sand.
So Galileo came to the formulation of the principle of inertia, according to which a material body maintains a state of rest or uniform rectilinear motion if no external forces act on it. This property of matter is often called inertia, and the movement of a body without external influences is called motion by inertia.
4. Newton's first law. In 1687, based on Galileo's principle of inertia, Newton formulated the first law of dynamics - Newton's first law:
A material point (body) is in a state of rest or uniform linear motion if other bodies do not act on it, or the forces acting from other bodies are balanced, i.e. compensated.
5.Free material point- a material point that is not affected by other bodies. Sometimes they say - an isolated material point.
6. Inertial reference system (IRS)– a reference system relative to which an isolated material point moves rectilinearly and uniformly, or is at rest.
Any reference system that moves uniformly and rectilinearly relative to the ISO is inertial,
Let us give another formulation of Newton's first law: There are reference systems relative to which a free material point moves rectilinearly and uniformly, or is at rest. Such reference systems are called inertial. Newton's first law is often called the law of inertia.
Newton's first law can also be given the following formulation: every material body resists a change in its speed. This property of matter is called inertia.
We encounter manifestations of this law every day in urban transport. When the bus suddenly picks up speed, we are pressed against the back of the seat. When the bus slows down, our body skids in the direction of the bus.
7. Non-inertial reference system – a reference system that moves unevenly relative to the ISO.
A body that, relative to the ISO, is in a state of rest or uniform linear motion. It moves unevenly relative to a non-inertial reference frame.
Any rotating reference system is a non-inertial reference system, because in this system the body experiences centripetal acceleration.
There are no bodies in nature or technology that could serve as ISOs. For example, the Earth rotates around its axis and any body on its surface experiences centripetal acceleration. However, for fairly short periods of time, the reference system associated with the Earth’s surface can, to some approximation, be considered ISO.
8.Galileo's principle of relativity. ISO can be as much salt as you like. Therefore, the question arises: what do the same mechanical phenomena look like in different ISOs? Is it possible, using mechanical phenomena, to detect the movement of the ISO in which they are observed.
The answer to these questions is given by the principle of relativity of classical mechanics, discovered by Galileo.
The meaning of the principle of relativity of classical mechanics is the statement: all mechanical phenomena proceed exactly the same way in all inertial frames of reference.
This principle can be formulated as follows: all laws of classical mechanics are expressed by the same mathematical formulas. In other words, no mechanical experiments will help us detect the movement of the ISO. This means that trying to detect ISO movement is meaningless.
We encountered the manifestation of the principle of relativity while traveling on trains. At the moment when our train is standing at the station, and the train standing on the adjacent track slowly begins to move, then in the first moments it seems to us that our train is moving. But it also happens the other way around, when our train smoothly picks up speed, it seems to us that the neighboring train has started moving.
In the above example, the principle of relativity manifests itself over small time intervals. As the speed increases, we begin to feel shocks and swaying of the car, i.e. our reference system becomes non-inertial.
So, trying to detect ISO movement is pointless. Consequently, it is absolutely indifferent which ISO is considered stationary and which is moving.
9. Galilean transformations. Let two ISOs move relative to each other with a speed. In accordance with the principle of relativity, we can assume that the ISO K is stationary, and the ISO moves relatively at a speed. For simplicity, we assume that the corresponding coordinate axes of the systems and are parallel, and the axes and coincide. Let the systems coincide at the moment of beginning and the movement occurs along the axes and , i.e. (Fig.28)
